Lecture 8: (Predicate) First Order Logic CS 580 (001) - Spring 2016

Amarda Shehu Department of Computer Science George Mason University, Fairfax, VA, USA

March 16, 2016

Amarda Shehu (580)

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Outline of Today’s Class

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Why First Order Logic (FOL)?

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FOL Syntax and Semantics

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Fun with Sentences

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Wumpus World in FOL

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FOL Summary

Amarda Shehu (580)

Outline of Today’s Class

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Pros and Cons of Propositional Logic

Propositional logic is declarative: pieces of syntax correspond to facts Propositional logic allows partial/disjunctive/negated information (unlike most data structures and databases) Propositional logic is compositional: meaning of B1,1 ∧ P1,2 is derived from meaning of B1,1 and of P1,2 Meaning in propositional logic is context-independent (unlike natural language, where meaning depends on context) Propositional logic has very limited expressive power (unlike natural language) E.g., cannot say “pits cause breezes in adjacent squares” except by writing one sentence for each square

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Why First Order Logic (FOL)?

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First-order Logic

Whereas propositional logic assumes world contains facts, first-order logic (like natural language) assumes the world contains Objects: people, houses, numbers, theories, Ronald McDonald, colors, baseball games, wars, centuries . . . Relations: red, round, bogus, prime, multistoried . . ., brother of, bigger than, inside, part of, has color, occurred after, owns, comes between, . . . Functions: father of, best friend, third inning of, one more than, end of . . .

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Why First Order Logic (FOL)?

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Logics in General

Language Propositional logic First-order logic Temporal logic Probability theory Fuzzy logic

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Ontological Commitment facts facts, objects, relations facts, objects, relations, times facts facts + degree of truth

Why First Order Logic (FOL)?

Epistemological Commitment true/false/unknown true/false/unknown true/false/unknown degree of belief known interval value

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Syntax of FOL: Basic Elements

Constants

KingJohn, 2, UCB, . . .

Predicates

Brother , >, . . .

Functions

Sqrt, LeftLegOf , . . .

Variables

x, y , a, b, . . .

Connectives

∧ ∨ ¬ =⇒

Equality

=

Quantifiers

∀∃

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⇔

FOL Syntax and Semantics

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Atomic Sentences

Atomic sentence

=

predicate(term1 , . . . , termn ) or term1 = term2

Term

=

function(term1 , . . . , termn ) or constant or variable

E.g.,

Brother (KingJohn, RichardTheLionheart) > (Length(LeftLegOf (Richard)), Length(LeftLegOf (KingJohn)))

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FOL Syntax and Semantics

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Complex Sentences

Complex sentences are made from atomic sentences using connectives ¬S,

E.g.

S1 ∧ S2 ,

S1 ∨ S2 ,

S1 =⇒ S2 ,

S1 ⇔ S2

Sibling (KingJohn, Richard) =⇒ Sibling (Richard, KingJohn) >(1, 2) ∨ ≤(1, 2) >(1, 2) ∧ ¬>(1, 2)

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FOL Syntax and Semantics

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Truth in First-order Logic

Sentences are true with respect to a model and an interpretation Model contains ≥ 1 objects (domain elements) and relations among them Interpretation specifies referents for constant symbols → objects predicate symbols → relations function symbols → functional relations An atomic sentence predicate(term1 , . . . , termn ) is true iff the objects referred to by term1 , . . . , termn are in the relation referred to by predicate

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FOL Syntax and Semantics

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Models for FOL: Example

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FOL Syntax and Semantics

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Truth Example

Consider the interpretation in which: Richard → Richard the Lionheart John → the evil King John Brother → the brotherhood relation Under this interpretation, Brother (Richard, John) is true just in case Richard the Lionheart and the evil King John are in the brotherhood relation in the model

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FOL Syntax and Semantics

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Models for FOL: Lots!

Entailment in propositional logic can be computed by enumerating models We can enumerate the FOL models for a given KB vocabulary: For each number of domain elements n from 1 to ∞ For each k-ary predicate Pk in the vocabulary For each possible k-ary relation on n objects For each constant symbol C in the vocabulary For each choice of referent for C from n objects . . . Computing entailment by enumerating FOL models is not easy!

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FOL Syntax and Semantics

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Universal Quantification

∀ hvariablesihsentencei Everyone at Berkeley is smart: ∀ xAt(x, Berkeley ) =⇒ Smart(x) ∀ xP is true in a model m iff P is true with x being each possible object in the model Roughly speaking, equivalent to the conjunction of instantiations of P ∧ ∧ ∧

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(At(KingJohn, Berkeley ) =⇒ Smart(KingJohn)) (At(Richard, Berkeley ) =⇒ Smart(Richard)) (At(Berkeley , Berkeley ) =⇒ Smart(Berkeley )) ...

FOL Syntax and Semantics

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A common Mistake to Avoid

Typically, ⇒ is the main connective with ∀ Common mistake: using ∧ as the main connective with ∀: ∀ x At(x,Berkeley) ∧ Smart(x) means “Everyone is at Berkeley and everyone is smart”

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FOL Syntax and Semantics

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Existential Quantification

∃ hvariablesihsentencei Someone at Stanford is smart: ∃ x At(x, Stanford) ∧ Smart(x) ∃ x P is true in a model m iff P is true with x being some possible object in the model Roughly speaking, equivalent to the disjunction of instantiations of P ∨ ∨ ∨

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(At(KingJohn, Stanford) ∧ Smart(KingJohn)) (At(Richard, Stanford) ∧ Smart(Richard)) (At(Stanford, Stanford) ∧ Smart(Stanford)) ...

FOL Syntax and Semantics

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Another Common Mistake to Avoid

Typically, ∧ is the main connective with ∃ Common mistake: using =⇒ as the main connective with ∃: ∃ x At(x,Stanford) =⇒ Smart(x) is true if there is anyone who is not at Stanford!

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x

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FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??)

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as ∃ y ∃ x

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as ∃ y ∃ x (why??)

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as ∃ y ∃ x (why??) ∃ x ∀ y

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as ∃ y ∃ x (why??) ∃ x ∀ y is not the same as ∀ y ∃ x

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as ∃ y ∃ x (why??) ∃ x ∀ y is not the same as ∀ y ∃ x ∃ x ∀ y Loves(x, y )

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as ∃ y ∃ x (why??) ∃ x ∀ y is not the same as ∀ y ∃ x ∃ x ∀ y Loves(x, y ) “There is a person who loves everyone in the world”

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as ∃ y ∃ x (why??) ∃ x ∀ y is not the same as ∀ y ∃ x ∃ x ∀ y Loves(x, y ) “There is a person who loves everyone in the world” ∀ y ∃ x Loves(x, y )

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as ∃ y ∃ x (why??) ∃ x ∀ y is not the same as ∀ y ∃ x ∃ x ∀ y Loves(x, y ) “There is a person who loves everyone in the world” ∀ y ∃ x Loves(x, y ) “Everyone in the world is loved by at least one person” Quantifier duality: each can be expressed using the other ∀ x Likes(x, IceCream) ¬ ∃ x ¬ Likes(x, IceCream) ∃ x Likes(x, Broccoli) ¬ ∀ x ¬ Likes(x, Broccoli)

Amarda Shehu (580)

FOL Syntax and Semantics

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Properties of Quantifiers

∀ x ∀ y is the same as ∀ y ∀ x (why??) ∃ x ∃ y is the same as ∃ y ∃ x (why??) ∃ x ∀ y is not the same as ∀ y ∃ x ∃ x ∀ y Loves(x, y ) “There is a person who loves everyone in the world” ∀ y ∃ x Loves(x, y ) “Everyone in the world is loved by at least one person” Quantifier duality: each can be expressed using the other ∀ x Likes(x, IceCream) ¬ ∃ x ¬ Likes(x, IceCream) ∃ x Likes(x, Broccoli) ¬ ∀ x ¬ Likes(x, Broccoli)

Amarda Shehu (580)

FOL Syntax and Semantics

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Fun with Sentences Brothers are siblings

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Fun with Sentences

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Fun with Sentences

Brothers are siblings

Amarda Shehu (580)

Fun with Sentences

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Fun with Sentences

Brothers are siblings ∀ x, y Brother (x, y ) =⇒ Sibling (x, y )

Amarda Shehu (580)

Fun with Sentences

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Fun with Sentences

Brothers are siblings ∀ x, y Brother (x, y ) =⇒ Sibling (x, y ) “Sibling” is symmetric

Amarda Shehu (580)

Fun with Sentences

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Fun with Sentences

Brothers are siblings ∀ x, y Brother (x, y ) =⇒ Sibling (x, y ) “Sibling” is symmetric ∀ x, y Sibling (x, y ) ⇔ Sibling (y , x)

Amarda Shehu (580)

Fun with Sentences

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Fun with Sentences

Brothers are siblings ∀ x, y Brother (x, y ) =⇒ Sibling (x, y ) “Sibling” is symmetric ∀ x, y Sibling (x, y ) ⇔ Sibling (y , x) One’s mother is one’s female parent

Amarda Shehu (580)

Fun with Sentences

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Fun with Sentences

Brothers are siblings ∀ x, y Brother (x, y ) =⇒ Sibling (x, y ) “Sibling” is symmetric ∀ x, y Sibling (x, y ) ⇔ Sibling (y , x) One’s mother is one’s female parent ∀ x, y Mother (x, y ) ⇔ (Female(x) ∧ Parent(x, y ))

Amarda Shehu (580)

Fun with Sentences

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Fun with Sentences

Brothers are siblings ∀ x, y Brother (x, y ) =⇒ Sibling (x, y ) “Sibling” is symmetric ∀ x, y Sibling (x, y ) ⇔ Sibling (y , x) One’s mother is one’s female parent ∀ x, y Mother (x, y ) ⇔ (Female(x) ∧ Parent(x, y )) A first cousin is a child of a parent’s sibling

Amarda Shehu (580)

Fun with Sentences

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Fun with Sentences

Brothers are siblings ∀ x, y Brother (x, y ) =⇒ Sibling (x, y ) “Sibling” is symmetric ∀ x, y Sibling (x, y ) ⇔ Sibling (y , x) One’s mother is one’s female parent ∀ x, y Mother (x, y ) ⇔ (Female(x) ∧ Parent(x, y )) A first cousin is a child of a parent’s sibling ∀ x, y FirstCousin(x, y ) ⇔ ∃ p, ps Parent(p, x) ∧ Sibling (ps, p) ∧ Parent(ps, y )

Amarda Shehu (580)

Fun with Sentences

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Fun with Sentences

Brothers are siblings ∀ x, y Brother (x, y ) =⇒ Sibling (x, y ) “Sibling” is symmetric ∀ x, y Sibling (x, y ) ⇔ Sibling (y , x) One’s mother is one’s female parent ∀ x, y Mother (x, y ) ⇔ (Female(x) ∧ Parent(x, y )) A first cousin is a child of a parent’s sibling ∀ x, y FirstCousin(x, y ) ⇔ ∃ p, ps Parent(p, x) ∧ Sibling (ps, p) ∧ Parent(ps, y )

Amarda Shehu (580)

Fun with Sentences

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Equality

term1 = term2 is true under a given interpretation if and only if term1 and term2 refer to the same object E.g.,

1 = 2 and ∀ x ×(Sqrt(x), Sqrt(x)) = x are satisfiable 2 = 2 is valid

E.g., definition of (full) Sibling in terms of Parent: ∀ x, y Sibling (x, y ) ⇔ [¬(x = y )∧ ∃ m, f ¬(m = f ) ∧ Parent(m, x) ∧ Parent(f , x) ∧ Parent(m, y ) ∧ Parent(f , y )]

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Fun with Sentences

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Interacting with FOL KBs

Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t = 5: Tell(KB, Percept([Smell, Breeze, None], 5)) Ask(KB, ∃ a Action(a, 5)) I.e., does KB entail any particular actions at t = 5?

Amarda Shehu (580)

Fun with Sentences

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Interacting with FOL KBs

Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t = 5: Tell(KB, Percept([Smell, Breeze, None], 5)) Ask(KB, ∃ a Action(a, 5)) I.e., does KB entail any particular actions at t = 5? Answer: Yes, {a/Shoot}

← substitution (binding list)

Given a sentence S and a substitution σ, Sσ denotes the result of plugging σ into S; e.g., S = Smarter (x, y ) σ = {x/Hillary , y /Bill} Sσ = Smarter (Hillary , Bill) Ask(KB, S) returns some/all σ such that KB |= Sσ

Amarda Shehu (580)

Fun with Sentences

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Interacting with FOL KBs

Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t = 5: Tell(KB, Percept([Smell, Breeze, None], 5)) Ask(KB, ∃ a Action(a, 5)) I.e., does KB entail any particular actions at t = 5? Answer: Yes, {a/Shoot}

← substitution (binding list)

Given a sentence S and a substitution σ, Sσ denotes the result of plugging σ into S; e.g., S = Smarter (x, y ) σ = {x/Hillary , y /Bill} Sσ = Smarter (Hillary , Bill) Ask(KB, S) returns some/all σ such that KB |= Sσ

Amarda Shehu (580)

Fun with Sentences

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Knowledge Base for the Wumpus World

“Perception” ∀ b, g , t Percept([Smell, b, g ], t) =⇒ Smelt(t) ∀ s, b, t Percept([s, b, Glitter ], t) =⇒ AtGold(t) Reflex: ∀ t AtGold(t) =⇒ Action(Grab, t) Reflex with internal state: do we have the gold already? ∀ t AtGold(t) ∧ ¬Holding (Gold, t) =⇒ Action(Grab, t) Holding (Gold, t) cannot be observed

⇒ keeping track of change is essential

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Wumpus World in FOL

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Deducing Hidden Properties Properties of locations: ∀ x, t At(Agent, x, t) ∧ Smelt(t) =⇒ Smelly (x) ∀ x, t At(Agent, x, t) ∧ Breeze(t) =⇒ Breezy (x) Squares are breezy near a pit: Diagnostic rule—infer cause from effect ∀ y Breezy (y ) =⇒ ∃ xPit(x) ∧ Adjacent(x, y ) Causal rule—infer effect from cause ∀ x, y Pit(x) ∧ Adjacent(x, y ) =⇒ Breezy (y ) Neither of these is complete—e.g., the causal rule doesn’t say whether squares far away from pits can be breezy Definition for the Breezy predicate: ∀ y Breezy (y ) ⇔ [∃ xPit(x) ∧ Adjacent(x, y )]

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Wumpus World in FOL

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Keeping Track of Change Facts hold in situations, rather than eternally E.g., Holding (Gold, Now ) rather than just Holding (Gold) Situation calculus is one way to represent change in FOL: Adds a situation argument to each non-eternal predicate E.g., Now in Holding (Gold, Now ) denotes a situation Situations are connected by the Result function Result(a, s) is the situation that results from doing a in s

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Wumpus World in FOL

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Describing Actions I

“Effect” axiom—describe changes due to action ∀ s AtGold(s) =⇒ Holding (Gold, Result(Grab, s)) “Frame” axiom—describe non-changes due to action ∀ s HaveArrow (s) =⇒ HaveArrow (Result(Grab, s)) Frame problem: find an elegant way to handle non-change (a) representation—avoid frame axioms (b) inference—avoid repeated “copy-overs” to keep track of state Qualification problem: true descriptions of real actions require endless caveats—what if gold is slippery or nailed down or . . . Ramification problem: real actions have many secondary consequences—what about the dust on the gold, wear and tear on gloves, . . .

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Wumpus World in FOL

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Describing Actions II

Successor-state axioms solve the representational frame problem Each axiom is “about” a predicate (not an action per se): P true afterwards

⇔

[an action made P true

∨

P true already and no action made P false]

For holding the gold: ∀ a, s Holding (Gold, Result(a, s)) ⇔ [(a = Grab ∧ AtGold(s)) ∨ (Holding (Gold, s) ∧ a 6= Release)]

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Wumpus World in FOL

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Making Plans

Initial condition in KB: At(Agent, [1, 1], S0 ) At(Gold, [1, 2], S0 ) Query: Ask(KB,∃ s Holding (Gold, s)) i.e., in what situation will I be holding the gold? Answer: {s/Result(Grab, Result(Forward, S0 ))} i.e., go forward and then grab the gold This assumes that the agent is interested in plans starting at S0 and that S0 is the only situation described in the KB

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Wumpus World in FOL

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Making Plans: A better way

Represent plans as action sequences [a1 , a2 , . . . , an ] PlanResult(p, s) is the result of executing p in s Then the query Ask(KB,∃ p Holding (Gold, PlanResult(p, S0 ))) has the solution {p/[Forward, Grab]} Definition of PlanResult in terms of Result: ∀ sPlanResult([ ], s) = s ∀ a, p, s PlanResult([a|p], s) = PlanResult(p, Result(a, s)) Planning systems are special-purpose reasoners designed to do this type of inference more efficiently than a general-purpose reasoner

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Wumpus World in FOL

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FOL Summary

First-order logic: – objects and relations are semantic primitives – syntax: constants, functions, predicates, equality, quantifiers Increased expressive power: sufficient to define wumpus world Situation calculus: – conventions for describing actions and change in FOL – can formulate planning as inference on a situation calculus KB

Amarda Shehu (580)

FOL Summary

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